It is well recognized that most rock formations in the Earth are to some extent anisotropic. For the most part, the anisotropy has been ignored in seismic processing. Its effects are small when compared to assumptions such as plane-layer approximation imposed by NMO and stack processing or the 2D earth model approximation imposed by 2D processing. However, with recent advances in inverse problem theory and supercomputing technology, most of these approximations have been overcome, leading to the development of 3D prestack imaging and inversion. Recent studies suggest that the type of details and accuracy expected from 3D prestack imaging and inversion can be distorted by neglecting anisotropy.
To include anisotropy in 3D prestack imaging and inversion, we need an accurate estimation of the anisotropic background velocity. The reconstruction of anisotropic background velocity models is probably the most important challenge in processing with anisotropy. Conventional methods for reconstructing the background velocity model, like migration-velocity methods, often assume an isotropic subsurface and can yield to inaccurate reservoir descriptions when the subsurface contains anisotropic rock formations.
The known migration-velocity method consists of scanning over different velocity models using prestack migration, the "good" velocity model is then constructed based on focusing analysis. Further details of this method are described for example by O. Yilmaz, Seismic Data Processing, Society of Exploration Geophysists, 1987.
An inversion method for data in the space-time (x-t) domain has for example been described by A. Tarantola in: Geophysical Prospecting, No. 32 (1984), 998-1015).
Another inversion process for data in the wavenumber-frequency (.omega.-k) domain has for example been described by L. Ikelle et al. in: Geophysics, Vol. 51, No. 6 (June 1986), 1266-1276.
An inversion method for isotropic media, also including a background velocity reconstruction, is described by L. Ikelle in: Geophys.J.Int., No. 123(1995), 507-528.
An attempt to reconstruct a velocity model for a transversely isotropic medium is described by T. A. Alkhalifah and I. Tsvankin in: 64th Ann. Intern. Mtg., Soc. Expl. Geophys. (1994), Expanded Abstracts, 1000-1003.
One reason why the isotropic linearized inversion or migration in the (.omega.-k) domain are widely used through the industry for reconstructing background velocity models is that the dispersion relationship which controls the extrapolation operation is explicit. This aspect has been extensively exploited over the last two decades to produce efficient migration and linearized inversion algorithms in the (.omega.-k) domain. For anisotropic media, the dispersion relationship is no longer explicit, it requires numerical resolution of an eigenvalue-eigenvector system for each point in the (.omega.-k) domain. Such computation is very slow. It renders inversion and migration, therefore anisotropic background reconstruction, economically unattractive.
In view of the above cited prior art it is an object of the invention to improve the known inversion methods for seismic data such as to include anisotropy in the subsurface formation.